d/(d{y)}(1/(sqrt({x)^2+{y)^2+{z}^2}})

Lösung

\frac{d}{d y} (\frac{1}{x ^{2} + y ^{2} + z ^{2}})

- \frac{y}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

Schritte zur Lösung

\frac{d}{d y} (\frac{1}{x ^{2} + y ^{2} + z ^{2}})

Wende Exponentenregel an:

\frac{1}{a} = a^{- 1}

= \frac{d}{d y} ((x^{2} + y^{2} + z^{2})^{- \frac{1}{2}})

Wende die Kettenregel an:

- \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \frac{d}{d y} (x^{2} + y^{2} + z^{2})

= - \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \frac{d}{d y} (x^{2} + y^{2} + z^{2})

\frac{d}{d y} (x^{2} + y^{2} + z^{2}) = 2 y

= - \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \cdot 2 y

Vereinfache

- \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \cdot 2 y : - \frac{y}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

= - \frac{y}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

\int - 2 ln (x) d x

\frac{d}{d m} (c^{im})

\frac{d}{d x} (5 x^{3} + 2 x^{2} + 6 x + 10)

d er i v a t i v e g (x) = \frac{1}{5 x ^{2} + 3 x}

x \to \infty lim (x^{- 7})